Learning objectives
The aim is to provide the basic instruments for the valuation of financial derivatives. at the end of the course, the student will know the basic principles of arbitrage pricing and completeness in the market, and the mathematical modeling of an elementary market model. In addition, the student will be familar with the representation of preferences for a rational decision maker and the optimal selection of a portfolio, given the returns and covariances of the traded assets.
Prerequisites
Basic elements of calculus and financial mathematics
Course unit content
Introduction to probability theory: the various approaches. The axiomatic approach. conditional probability and Bayes'theorem. Random numbers: the discrete case and the continuous case. Random vectors. Basic notions on financial markets. One-period financial market. Fundamental theorems of asset pricing. Pricing of derivatives. Introduction to expected utility theory. Portfolio selection: Markowitz's model.
Full programme
Introduction to probability theory. Classical, empirical and subjective approaches. Axiomatic approach: sample space, sigma-algebra and probability measure. Axioms of probability. conditional probability, Bayes theorem. Random numbers, measurability. Distribution function. Discrete random numbers: probability mass function. Continuous random numbers: density function.
Expectation, variance and standard deviation. Moments of a random number.
Random vectors. Independent random numbers. Covariance and correlation.
Introduction to financial market. A 1-period financial market, with zero e non-zero interest rate.
Law of one price. Arbitrage and completeness. State price densities and risk-neutral probabilities. Fundamental theorems of asset pricing. Derivatives: call and put options. Put-call parity. Forward contracts and forward prices.
Introduction to expected utility theory. Von-Neumann-Morgenstern axioms. Expected Utility theorem. Portfolio selection: Mean-variance principle. Markowitz's model.
Bibliography
E. CASTAGNOLI, Brevissimo Abbecedario di Matematica Finanziaria, scaricabile dalla sezione "materiali didattici" o disponibile presso il Centro fotocopie della Facoltà.
E. CASTAGNOLI, M. CIGOLA, L. PECCATI, Probability. A Brief Introduction, 2° edizione, Egea, 2009
Teaching methods
Oral lectures.
Assessment methods and criteria
Written exam.
Other information
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